Given a relative faithfully flat pointed scheme over the spectrum of a discrete valuation ring $X\rightarrow S$ , this paper is motivated by the study of the natural morphism from the fundamental group scheme of the generic fiber $X_{\unicode[STIX]{x1D702}}$ to the generic fiber of the fundamental group scheme of $X$ . Given a torsor $T\rightarrow X_{\unicode[STIX]{x1D702}}$ under an affine group scheme $G$ over the generic fiber of $X$ , we address the question of finding a model of this torsor over $X$ , focusing in particular on the case where $G$ is finite. We provide several answers to this question, showing for instance that, when $X$ is integral and regular of relative dimension 1, such a model exists on some model $X^{\prime }$ of $X_{\unicode[STIX]{x1D702}}$ obtained by performing a finite number of Néron blowups along a closed subset of the special fiber of $X$ . Furthermore, we show that when $G$ is étale, then we can find a model of $T\rightarrow X_{\unicode[STIX]{x1D702}}$ under the action of some smooth group scheme. In the first part of the paper, we show that the relative fundamental group scheme of $X$ has an interpretation as the Tannaka Galois group of a Tannakian category constructed starting from the universal torsor.

A method of choice for realizing finite groups as regular Galois groups over $\mathbb{Q}(T)$ is to find $\mathbb{Q}$-rational points on Hurwitz moduli spaces of covers. In another direction, the use of the so-called patching techniques has led to the realization of all finite groups over $\mathbb{Q}_p(T)$. Our main result shows that, under some conditions, these $p$-adic realizations lie on some special irreducible components of Hurwitz spaces (the so-called Harbater–Mumford components), thus connecting the two main branches of the area. As an application, we construct, for every projective system $(G_n)_{n\geq0}$ of finite groups, a tower of corresponding Hurwitz spaces $(\mathcal{H}_{G_n})_{n\geq0}$, geometrically irreducible and defined over some cyclotomic extension of $\mathbb{Q}$, which admits projective systems of $\mathbb{Q}_p^{\mathrm{ur}}$-rational points for all primes $p$ not dividing the orders $|G_n|$ ($n\geq0$).